- INTRODUCTION (0.1) In complex analysis of several variables it has turned out that Stein spaces are the equivalent to non-compact Riemann surfaces in classic function theory. On a Stein space there exist "enough" holomorphic functions, i.e., the global holomorphic functions describe completely the local function theory of this space. More precisely: let be given the algebra r(X, P) of all holomorphic functions on a Stein space (X, 8), X the basic topological space and P the sheaf of germs of holomorphic functions on X, then both, X as well as P, can be rediscovered by r (X, Fp) uniquely up to biholomorphic maps. Namely, X turns out to be homeomorphic to the spectrum uT(X, @) and P can be obtained by a certain power series technique due to Forster [S].
We consider only reduced complex analytic spaces (X, 8). It is well known that T(X, P) becomes a uniform Frechet-nuclear algebra when endowed with the topology of compact convergence on X. A topological Calgebra XZ' is called a Stein algebra if there exists a Stein space (X, 8) such that & is topologically isomorphic to r(X, 6). In this paper we wish to characterize Stein algebras by "intrinsic properties," that is, by purely topoiogically algebraic properties which do not depend on the function theory of the corresponding Stein space. Thereby we achieve a reconstruction of holomorphy by functionalanalytic principles. (0.2) The simplest class of non-trivial Stein algebras is the one of onedimensional regular Stein algebras. Since they correspond with the class of non-compact Riemann surfaces they are called Riemann algebras in [ 111. A first and rather difficult characterization of Riemann algebras is due to Richards [ 141. A simple characterization has been gicen in [ 111 by using Gleason's famous theorem [7] and a theorem of Carpenter [3]. Unfortunately, these methods do not seem to be transferable to dimension >2. In order to introduce analytic structure into spectra we apply a theorem